Hindawi Discrete Dynamics in Nature and Society Volume 2017, Article ID 4149320, 8 pages https://doi.org/10.1155/2017/4149320
Research Article Arbitrary Order Fractional Difference Operators with Discrete Exponential Kernels and Applications Thabet Abdeljawad,1 Qasem M. Al-Mdallal,2 and Mohamed A. Hajji2 1
Department of Mathematics and Physical Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia Department of Mathematical Sciences, United Arab Emirates University, P.O. Box 17551, Al Ain, Abu Dhabi, UAE
2
Correspondence should be addressed to Thabet Abdeljawad;
[email protected] Received 3 April 2017; Accepted 25 May 2017; Published 21 June 2017 Academic Editor: Garyfalos Papashinopoulos Copyright ยฉ 2017 Thabet Abdeljawad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Recently, Abdeljawad and Baleanu have formulated and studied the discrete versions of the fractional operators of order 0 < ๐ผ โค 1 with exponential kernels initiated by Caputo-Fabrizio. In this paper, we extend the order of such fractional difference operators to arbitrary positive order. The extension is given to both left and right fractional differences and sums. Then, existence and uniqueness theorems for the Caputo (CFC) and Riemann (CFR) type initial difference value problems by using Banach contraction theorem are proved. Finally, a Lyapunov type inequality for the Riemann type fractional difference boundary value problems of order 2 < ๐ผ โค 3 is proved and the ordinary difference Lyapunov inequality then follows as ๐ผ tends to 2 from right. Illustrative examples are discussed and an application about Sturm-Liouville eigenvalue problem in the sense of this new fractional difference calculus is given.
1. Introduction In the last few decades, the continuous and discrete fractional differential equations have received considerable interest due to their importance in many scientific fields; see, by way of example not exhaustive enumeration, [1โ7]. In [8], the authors introduced a fractional derivative with an exponential kernel which tends to the ordinary derivative as ๐ผ tends to 1. More properties of this fractional derivative have been studied in [9], where the correspondent fractional integral operator was formulated. Then, the authors in [7] defined the left and right fractional derivatives with exponential kernel in the Riemann sense and formulated the right fractional derivatives in the sense of Caputo-Fabrizio with complete investigation to the correspondent fractional integrals and all the discrete versions with integration and summation by parts applied in the fractional and discrete fractional variational calculus. Then, very recently, the same authors proved an interesting monotonicity result in the sense of this new fractional difference calculus in [10]. In the same direction, for the purpose of providing more fractional derivatives with different nonsingular kernels, the authors in [11] defined a fractional operator with
Mittag-Leffler kernel and in [12, 13] the complete details and discrete versions have been studied. The exponential kernel fractional derivatives and hence their discrete counterparts are quite different from the Mittag-Leffler kernel fractional operators. For example, the integral operator corresponding to exponential kernel fractional derivatives consists of a multiple of the function ๐ added to a multiple of the integration of ๐, whereas the Mittag-Leffler kernel correspondent integral operator consists of a multiple of ๐ and a Riemann-Liouville fractional integral of the same order. Also, the monotonicity coefficient of the CFR fractional difference operator of order 0 < ๐ผ โค 1 is ๐ผ as shown in [10], whereas for the discrete Mittag-Leffler CFR operator is ๐ผ2 as proven in [14]. Motivated, by what we mentioned above, we extend the order of fractional difference type operators with discrete exponential kernels to arbitrary positive order, prove existence and uniqueness theorems for the fractional initial value difference problems, and finally prove a Lyapunov type inequality for the CFR fractional difference operators of order 2 < ๐ผ โค 3. The ordinary discrete Lyapunov inequality is then confirmed as ๐ผ tends to 2 from the right not as in the case of the classical fractional difference as ๐ผ tends to 2 from the left [15]. For various fractional Lyapunov extensions we refer, for
2
Discrete Dynamics in Nature and Society
example, to [16โ29]. All the authors there were motivated by the following theorem on ordinary Lyapunov inequality.
This is fractionalising of the ๐-iterated nabla sum
๓ธ ๓ธ
๐ฆ (๐ก) + ๐ (๐ก) ๐ฆ (๐ก) = 0,
๐ก โ (๐, ๐) , ๐ฆ (๐) = ๐ฆ (๐) = 0, (1)
has a nontrivial solution, where ๐ is a real continuous function; then ๐
4 ๓ตจ ๓ตจ โซ ๓ตจ๓ตจ๓ตจ๐ (๐ )๓ตจ๓ตจ๓ตจ ๐๐ > . ๐โ๐ ๐
Theorem 2 (see [26]). Let ๐(๐ก) be a nontrivial, continuous, and nonnegative function with period ๐ and let ๐
0
4 . ๐
๐ฅ (๐ก) + ๐ (๐ก) ๐ฅ (๐ก) = 0,
โ โ < ๐ก < โ,
1 ๐โ1 ๐ผโ1 โ (๐ โ ๐ (๐ก)) ๐ (๐ ) , ฮ (๐ผ) ๐ =๐ก
(โ๐โ๐ผ ๐) (๐ก) =
Definition 4 (see [7, 10]). For ๐ผ โ (0, 1) and ๐ defined on N๐ , or ๐ N = {๐, ๐ โ 1, . . .} in right case, we have the following definitions: (i) The left (nabla) CFC fractional difference is given by
(3) ๐ผ
(CFC๐ โ ๐) (๐ก) =
๐ต (๐ผ) ๐ก โ (โ ๐) (๐ ) (1 โ ๐ผ)๐กโ๐(๐ ) 1 โ ๐ผ ๐ =๐+1 ๐ ๐ก
= ๐ต (๐ผ) โ (โ๐ ๐) (๐ ) (1 โ ๐ผ)
(4)
For the classical fractional calculus which is behind many extensions, we refer the reader to [32โ35] and for the sake of comparison with the classical discrete fractional case we refer to [36] and the references therein. In addition, for the discrete fractional operators and their duality we refer to [37โ39]. The article will be organised as follows: In the remaining part of this section we shall give some basics about the discrete CFC and CFR fractional differences and their correspondent sums as used in [7, 10]. In Section 2, we extend the order of CFC and CFR fractional differences and their correspondent sums to arbitrary positive order and give some illustrative examples. In Section 3, we prove some existence and uniqueness theorems by means of Banach fixed point theorem and give some illustrative examples. In Section 4, we prove a Lyapunov type inequality for a fractional CFR difference boundary value problem of order 2 < ๐ผ โค 3 and give an application to the fractional difference Sturm-Liouville Eigenvalue problem (SLEP) to enrich the applicability of our proven Lyapunov inequality in the frame of fractional difference operators with discrete exponential kernels.
(8)
.
(ii) The right (nabla) CFC fractional difference has the following form: ๐ผ
(CFC โ๐ ๐) (๐ก) =
(5)
(9)
= ๐ต (๐ผ) โ (โฮ ๐ ๐) (๐ ) (1 โ ๐ผ)๐ โ๐ก . ๐ =๐ก
(iii) The left (nabla) CFR fractional difference is written as ๐ผ
(CFR๐ โ ๐) (๐ก) =
๐ก ๐ต (๐ผ) โ๐ก โ ๐ (๐ ) (1 โ ๐ผ)๐กโ๐(๐ ) 1 โ ๐ผ ๐ =๐+1 ๐ก
= ๐ต (๐ผ) โ๐ก โ ๐ (๐ ) (1 โ ๐ผ)
๐กโ๐
(10)
.
๐ =๐+1
(iv) The right (nabla) CFR fractional difference is given by ๐ผ
Definition 3 (see [36]). For ๐ผ > 0, ๐ โ R, ๐(๐ ) = ๐ โ 1, and ๐ a real-valued function defined on N๐ = {๐, ๐ + 1, . . .}, the left Riemann-Liouville fractional sum of order ๐ผ > 0 is defined by
๐ต (๐ผ) ๐โ1 โ (โฮ ๐ ๐) (๐ ) (1 โ ๐ผ)๐ โ๐(๐ก) 1 โ ๐ผ ๐ =๐ก ๐โ1
(CFR โ๐ ๐) (๐ก) =
2. Preliminaries
๐ก 1 ๐ผโ1 โ (๐ก โ ๐ (๐ )) ๐ (๐ ) . ฮ (๐ผ) ๐ =๐+1
๐กโ๐
๐ =๐+1
are purely imaginary with modulus one.
(๐ โโ๐ผ ๐) (๐ก) =
(7)
where ๐ก๐ผ = ฮ(๐ก + ๐ผ)/ฮ(๐ก) and ฮ(๐ก) is the well-known gamma special function of ๐ก.
Then the roots of the characteristic equation corresponding to Hills equation ๓ธ ๓ธ
(6)
The right fractional integral ending at ๐, where usually we assume that ๐ โก ๐ (mod 1), is defined by
(2)
Notice that inequality (2) is known as the classical Lyapunov inequality. It is worth mentioning that Cheng [31] had pointed out that Lyapunov neither stated nor proved Theorem 1 but he only stated the following result.
โซ ๐ (๐ ) ๐๐ โค
๐ก 1 ๐โ1 โ (๐ก โ ๐ (๐ )) ๐ (๐ ) . (๐ โ 1)! ๐+1
(๐ โโ๐ ๐) (๐ก) =
Theorem 1 (see [30]). If the boundary value problem
๐โ1 ๐ต (๐ผ) (โฮ ๐ก ) โ ๐ (๐ ) (1 โ ๐ผ)๐ โ๐(๐ก) 1โ๐ผ ๐ =๐ก ๐โ1
(11)
= ๐ต (๐ผ) (โฮ ๐ก ) โ ๐ (๐ ) (1 โ ๐ผ)๐ โ๐ก , ๐ =๐ก
where ๐ต(๐ผ) is a normalization positive constant depending on ๐ผ satisfying ๐ต(0) = ๐ต(1) = 1, (โ๐)(๐ก) = ๐(๐ก) โ ๐(๐ก โ 1), and (ฮ๐)(๐ก) = ๐(๐ก + 1) โ ๐(๐ก).
Discrete Dynamics in Nature and Society โ๐ผ
3 ๐ผ
In [7, 8], it was verified that (CF๐ โ CFR๐ โ ๐)(๐ก) = ๐(๐ก) ๐ผ โ๐ผ and (CFR๐ โ CF๐ โ ๐)(๐ก) = ๐(๐ก). Also, in the right case ๐ผ ๐ผ ๐ผ โ๐ผ (CF ๐ผ๐ CFR โ๐ ๐)(๐ก) = ๐(๐ก) and (CFR โ๐ CF โ๐ ๐)(๐ก) = ๐(๐ก). From [7, 8] we recall the relation between the CFC and CFR fractional differences as ๐ผ
๐ผ
(CFC๐ โ ๐) (๐ก) = (CFR๐ โ ๐) (๐ก) โ
๐ต (๐ผ) ๐ (๐) (1 โ ๐ผ)๐กโ๐ , 1โ๐ผ
(12)
โ1
noting that (CF๐ โ ๐)(๐ก) = (๐ โโ1 ๐)(๐ก), we see that for ๐ผ = ๐+1 โ๐ผ we have (CF๐ โ ๐)(๐ก) = (๐ โโ(๐+1) ๐)(๐ก). Also, for 0 < ๐ผ โค 1 we reobtain the concepts defined in Definition 4. Therefore, our generalization to higher order case is confirmed. Analogously, in the right case we have the following extension. Definition 8. Let ๐ < ๐ผ โค ๐ + 1 and ๐ be defined on N๐ โฉ ๐ N. Set ๐ฝ = ๐ผ โ ๐. Then ๐ฝ โ (0, 1] and we define ๐ผ
and for the right case by ๐ผ
๐ผ
(CFC โ๐ ๐) (๐ก) = (CFR โ๐ ๐) (๐ก) ๐ต (๐ผ) ๐ (๐) (1 โ ๐ผ)๐โ๐ก . โ 1โ๐ผ
๐ผ (CFR โ๐ ๐) (๐ก)
(13)
Lemma 5 (see [7]). For 0 < ๐ผ < 1, we have โ๐ผ ๐ถ๐น๐ถ ๐ผ ๐ โ ๐) (๐ก)
โ๐ผ ๐ผ (๐ถ๐น โ๐ ๐ถ๐น๐ถโ๐ ๐) (๐ก)
โ๐ผ
= ๐ (๐ก) โ ๐ (๐) , (14) = ๐ (๐ก) โ ๐ (๐) .
Proposition 9. For ๐ defined on N๐ โฉ ๐ N and ๐ < ๐ผ โค ๐ + 1, we have ๐ผ
๐ผ
๐ times
(iii) (โ ฮ ๐)(๐ก) = (โ1) (ฮ ๐)(๐ก).
๐ผ
๐ผ
(๐ถ๐น๐ถโ๐ ๐) (๐ก) = (๐ถ๐น๐
โ๐ ๐) (๐ก)
3. Higher Order Fractional Differences and Sums
โ
Definition 6. Let ๐ < ๐ผ โค ๐ + 1 and ๐ be defined on N๐ โฉ ๐ N. Set ๐ฝ = ๐ผ โ ๐ and define ๐ฝ
(CFC๐ โ ๐) (๐ก) = (CFC๐ โ โ๐ ๐) (๐ก) , ๐ฝ (CFR๐ โ โ๐ ๐) (๐ก) .
(15)
โ๐ฝ (๐ โโ๐ CF๐ โ ๐) (๐ก) .
๐ต (๐ผ) ( ฮ๐ ๐) (๐) (1 โ ๐ผ)๐โ๐ก . 1โ๐ผ โ
Proposition 10. For ๐ข(๐ก) defined on N๐ โฉ ๐ N and for some ๐ โ N0 with ๐ < ๐ผ โค ๐ + 1, we have (i) (๐ถ๐น๐
๐ โ
= ๐ข(๐ก).
โ๐ผ ๐ถ๐น๐
๐ผ ๐ โ ๐ข)(๐ก)
๐ ๐ = ๐ข(๐ก) โ โ๐โ1 ๐=0 ((โ ๐ข)(๐)/๐!)(๐ก โ ๐) .
โ๐ผ ๐ถ๐น๐ถ ๐ผ ๐ โ ๐ข)(๐ก)
= ๐ข(๐ก) โ โ๐๐=0 ((โ๐ ๐ข)(๐)/๐!)(๐ก โ ๐)๐ .
(16)
(ii) (๐ถ๐น๐ โ
Note that if we use the convention that (๐ โโ0 ๐)(๐ก) = ๐(๐ก), then for the case 0 < ๐ผ โค 1 we have ๐ฝ = ๐ผ and hence we obtain Definition 4. Also, the convention (โ0 ๐)(๐ก) = ๐(๐ก) ๐ผ ๐ผ leads to (CFR๐ โ ๐)(๐ก) and (CFC๐ โ ๐)(๐ก) as in Definition 4 for 0 < ๐ผ โค 1.
(iii) (๐ถ๐น๐ โ
Remark 7. In Definition 6, if we let ๐ผ = ๐ + 1 then ๐ฝ = 1 and ๐ผ 1 hence (CFR๐ โ ๐)(๐ก) = (CFR๐ โ โ๐ ๐)(๐ก) = (โ๐+1 ๐)(๐ก). Also, by
(20)
Next proposition explains the action of the arbitrary โ๐ผ order sum operator CF๐ โ on the arbitrary order CFR and CFC differences (and vice versa) and the action of the CFR difference on the CF correspondent sum operator.
๐ผ ๐ถ๐น โ๐ผ ๐ โ ๐ข)(๐ก)
The associated fractional sum is given by =
(19)
and for the right case
๐ times ๐ ๐
โ๐ผ (CF๐ โ ๐) (๐ก)
๐ต (๐ผ) ๐ (โ ๐) (๐) (1 โ ๐ผ)๐กโ๐ , 1โ๐ผ
โ
โโโโโโโโโโโโโโโ โ
โ
โ
ฮ๐)(๐ก). (ii) (ฮ ๐)(๐ก) = (ฮฮ
=
(18)
(๐ถ๐น๐ถ๐โ ๐) (๐ก) = (๐ถ๐น๐
๐ โ ๐) (๐ก)
๐
๐ผ (CFR๐ โ ๐) (๐ก)
โ๐ฝ
(CF โ๐ ๐) (๐ก) = (โ๐โ๐ CF โ๐ ๐) (๐ก) .
โโโโโโโโโโโโโโโ โ
โ
โ
โ๐)(๐ก). (i) (โ๐ ๐)(๐ก) = (โโ
๐ผ
(17)
An immediate extension of (12) and (13) by using Definition 6 is the following.
Notation. For a positive integer ๐, we have
๐
=
CFR ๐ฝ โ๐ (โ ฮ๐ ๐) (๐ก) .
The associated fractional integral is given by
Notice that we extend Definition 4 to arbitrary ๐ผ > 0 in the next section.
(๐ถ๐น๐ โ
๐ฝ
(CFC โ๐ ๐) (๐ก) = CFC โ๐ (โ ฮ๐ ๐) (๐ก) ,
Proof. (i) By Definition 6 and the statement after Definition 4, we have (CFR๐ โ
๐ผ CF โ๐ผ ๐ โ ๐ข) (๐ก)
๐ฝ
โ๐ฝ
= CFR๐ โ โ๐ ๐ โโ๐ CF๐ โ ๐ข (๐ก) =
CFR ๐ฝ CF โ๐ฝ ๐ โ ๐ โ ๐ข (๐ก)
= ๐ข (๐ก) .
(21)
4
Discrete Dynamics in Nature and Society Notice that the condition ๐น(๐) = 0 verifies the initial condition ๐ฆ(๐) = ๐. In addition, when ๐ผ โ 1 we obtain the solution of the ordinary difference initial value problem (โ๐ฆ)(๐ก) = ๐น(๐ก), ๐ฆ(๐) = ๐. (ii) Assume 1 < ๐ผ โค 2, ๐น(๐) = 0, ๐ฆ(๐) = ๐1 , and โ๐ผ (โ๐ฆ)(๐) = ๐2 . By applying CF๐ โ and making use of Proposition 10 and Definition 6 with ๐ฝ = ๐ผ โ 1, we obtain the solution
(ii) By Definition 6 and the statement after Definition 4, we have (CF๐ โ
โ๐ผ CFR ๐ผ ๐ โ ๐ข) (๐ก)
โ๐ฝ CFR ๐ฝ ๐ ๐ โ โ ๐ข) (๐ก)
= (๐ โโ๐ CF๐ โ
= ๐ โโ๐ โ๐ ๐ข (๐ก)
(22)
๐โ1 (โ๐ ๐ข) (๐)
= ๐ข (๐ก) โ โ
๐!
๐=0
(๐ก โ ๐)๐ ,
๐ฆ (๐ก) = ๐1 + ๐2 (๐ก โ ๐) +
where ๐ฝ = ๐ผ โ ๐. (iii) By Lemma 5 applied to ๐(๐ก) = (โ๐ ๐ข)(๐ก), we have (CF๐ โ
โ๐ผ CFC ๐ผ ๐ โ ๐ข) (๐ก)
๐ก ๐ผโ1 + โ (๐ก โ ๐ (๐ )) ๐น (๐ ) . ๐ต (๐ผ โ 1) ๐ =๐+1
๐ฝ
= ๐ โโ๐ ๐ โโ๐ฝCFC๐ โ โ๐ ๐ข (๐ก)
๐โ1 (โ๐ ๐ข) (๐)
๐!
๐=0
โ (โ๐ ๐ข) (๐) ๐
= ๐ข (๐ก) โ โ
(23)
(โ๐ ๐ข) (๐) ๐!
๐=0
(๐ก โ ๐)๐
(๐ก โ ๐)๐ ๐!
In the next section, we prove existence and uniqueness theorems for some types of CFC and CFR initial value difference problems.
(๐ก โ ๐)๐ .
Example 13. Consider the CFC difference boundary value problem ๐ผ
(CFC๐ โ ๐ฆ) (๐ก) + ๐ (๐ก) ๐ฆ (๐ก) = 0,
Using the facts that โ ฮ๐ โ๐โ๐ ๐(๐ก) = ๐(๐ก), ๐โ1 ( ฮ โ
โ๐๐ โ ฮ๐ ๐ (๐ก) = ๐ (๐ก) โ โ
๐=0
๐
๐) (๐)
๐!
(๐ โ ๐ก)๐ ,
1 < ๐ผ โค 2, ๐ก โ N๐,๐ , ๐ฆ (๐) = ๐ฆ (๐) = 0. (24)
Proposition 11. For ๐ข(๐ก) defined on N๐,๐ and ๐ผ โ (๐, ๐ + 1], for some ๐ โ N0 , we have
Then ๐ฝ = ๐ผ โ 1 and by Proposition 10 if we apply the operator we obtain the solution โ๐ผ
๐ฆ (๐ก) = ๐1 + ๐2 (๐ก โ ๐) โ (CF๐ โ ๐ (โ
) ๐ฆ (โ
)) (๐ก) . โ๐ผ
(CF๐ โ ๐ (โ
) ๐ฆ (โ
)) (๐ก) =
โ๐ผ
โ๐ผ ๐ถ๐น๐
๐ผ โ๐ ๐ข)(๐ก) โ๐ผ ๐ผ (๐ถ๐น โ๐ ๐ถ๐น๐ถโ๐ ๐ข)(๐ก)
(iii)
Example 12. Consider the initial value problem: ๐ผ
(CFC๐ โ ๐ฆ) (๐ก) = ๐น (๐ก) ,
(25)
(i) Assume 0 < ๐ผ โค 1, ๐ฆ(๐) = ๐, and ๐น(๐) = 0. By โ๐ผ applying CF๐ โ and making use of Proposition 10, we get the solution ๐ฆ (๐ก) = ๐ +
๐ก
๐ผ 1โ๐ผ ๐น (๐ก) + โ ๐น (๐ ) . ๐ต (๐ผ) ๐ต (๐ผ) ๐ =๐+1
(26)
(30)
Hence, the solution has the form ๐ฆ (๐ก) = ๐1 + ๐2 (๐ก โ ๐) โ
where ๐น(๐ก) is defined on N๐,๐ = N๐ โฉ ๐ N. Let us consider two cases depending on the order ๐ผ > 0:
1โ๐ฝ ๐ก โ ๐ (๐ ) ๐ฆ (๐ ) ๐ต (๐ฝ) ๐ =๐+1 ๐ฝ โโ2 ๐ (๐ก) ๐ฆ (๐ก) . + ๐ต (๐ฝ) ๐
๐ ๐ = ๐ข(๐ก)โโ๐โ1 ๐=0 ((โ ฮ ๐ข)(๐)/๐!)(๐โ๐ก) .
= ๐ข(๐ก)โโ๐๐=0 ((โ ฮ๐ ๐ข)(๐)/๐!)(๐โ๐ก)๐ .
(29)
But
(i) (๐ถ๐น๐
โ๐ ๐ถ๐น โ๐ ๐ข)(๐ก) = ๐ข(๐ก). (ii) (๐ถ๐น โ๐
(28)
CF โ๐ผ ๐โ ,
and making use of Lemma 5 and the statement after Definition 4, we can state, for the right case, the following.
๐ผ
(27)
Notice that the solution ๐ฆ(๐ก) verifies ๐ฆ(๐) = ๐1 without the use of ๐น(๐) = 0. However, it verifies (โ๐ฆ)(๐) = ๐2 under the assumption ๐น(๐) = 0. Also, note that when ๐ผ โ 2 we recover the solution of the second-order ordinary difference initial value problem (โ2 ๐ฆ)(๐ก) = ๐น(๐ก), ๐ฆ(๐) = ๐1 , and (โ๐ฆ)(๐) = ๐2 .
= ๐ โโ๐ [โ๐ ๐ข (๐ก) โ (โ๐ ๐ข) (๐)] = ๐ข (๐ก) โ โ
๐ก 2โ๐ผ โ ๐พ (๐ ) ๐ต (๐ผ โ 1) ๐ =๐+1
๐ก 2โ๐ผ โ ๐ (๐ ) ๐ฆ (๐ ) ๐ต (๐ผ โ 1) ๐ =๐+1
๐ก ๐ผโ1 โ โ (๐ก โ ๐ (๐ )) ๐ (๐ ) ๐ฆ (๐ ) . ๐ต (๐ผ โ 1) ๐ =๐+1
(31)
The boundary conditions imply that ๐1 = 0 and ๐2 =
๐ 2โ๐ผ โ ๐ (๐ ) ๐ฆ (๐ ) (๐ โ ๐) ๐ต (๐ผ โ 1) ๐ =๐+1 ๐ ๐ผโ1 + โ (๐ โ ๐ (๐ )) ๐ (๐ ) ๐ฆ (๐ ) . (๐ โ ๐) ๐ต (๐ผ โ 1) ๐ =๐+1
(32)
Discrete Dynamics in Nature and Society
5 Proposition 10. The condition ๐(๐, ๐ฆ(๐)) = 0 always can not be avoided as we have seen in Example 12 with ๐(๐ก, ๐ฆ(๐ก)) = ๐น(๐ก). As a result of Theorem 14, we conclude that the fractional difference linear initial value problem
Hence, ๐ฆ (๐ก) =
(2 โ ๐ผ) (๐ก โ ๐) ๐ โ ๐ (๐ ) ๐ฆ (๐ ) (๐ โ ๐) ๐ต (๐ผ โ 1) ๐ =๐+1 โ
(๐ผ โ 1) (๐ก โ ๐) โ (๐ โ ๐ (๐ )) ๐ (๐ ) ๐ฆ (๐ ) (๐ โ ๐) ๐ต (๐ผ โ 1) ๐ =๐+1
๐ก 2โ๐ผ โ โ ๐ (๐ ) ๐ฆ (๐ ) ๐ต (๐ผ โ 1) ๐ =๐+1
โ
๐ผ
(CFC๐ โ ๐ฆ) (๐ก) = ๐๐ฆ (๐ก) ,
๐
(33)
๐ก ๐ผโ1 โ (๐ก โ ๐ (๐ )) ๐ (๐ ) ๐ฆ (๐ ) . ๐ต (๐ผ โ 1) ๐ =๐+1
๐ โ R, ๐ก โ N๐,๐ , 0 < ๐ผ โค 1, ๐ฆ (๐) = ๐,
(38)
can have only the trivial solution unless ๐ผ = 1. Indeed, the solution satisfies ๐ฆ(๐ก) = ๐ + ๐((1 โ ๐ผ)/๐ต(๐ผ))๐ฆ(๐ก) + (๐ผ๐/๐ต(๐ผ)) โ๐ก๐ =๐+1 ๐ฆ(๐ ). This solution is only verified at ๐ if (1 โ ๐ผ)๐ฆ(๐) = 0. Theorem 16. Consider the system ๐ผ
(๐ถ๐น๐
๐ โ ๐ฆ) (๐ก) = ๐ (๐ก, ๐ฆ (๐ก)) ,
4. Existence and Uniqueness Theorems for the Initial Value Problem Types
๐ก โ N๐,๐ , 1 < ๐ผ โค 2, ๐ฆ (๐) = ๐,
In this section we prove existence and uniqueness theorems for CFC and CFR type initial value problems. Theorem 14. Consider the system
๐ก โ N๐,๐ , 0 < ๐ผ โค 1, ๐ฆ (๐) = ๐,
(34)
๐ถ๐น โ๐ผ ๐ โ ๐ (๐ก, ๐ฆ (๐ก)) .
(35)
Proof. First, by the help of Proposition 10, (12), and taking into account the fact that ๐(๐, ๐ฆ(๐)) = 0, it is straight forward to prove that ๐ฆ(๐ก) satisfies system (34) if and only if it satisfies (35). Let ๐ = {๐ฅ : max๐กโN๐,๐ |๐ฅ(๐ก)| < โ} be the Banach space endowed with the norm โ๐ฅโ = max๐กโN๐,๐ |๐ฅ(๐ก)|. On ๐ define the linear operator โ๐ผ
(๐๐ฅ) (๐ก) = ๐ + CF๐ โ ๐ (๐ก, ๐ฅ (๐ก)) .
๐ฆ (๐ก) = ๐ + ๐ถ๐น๐ โ ๐ (๐ก, ๐ฆ (๐ก)) =๐+
such that ๐(๐, ๐ฆ(๐)) = 0, ๐ด((1 โ ๐ผ)/๐ต(๐ผ) + ๐ผ(๐ โ ๐)/๐ต(๐ผ)) < 1, and |๐(๐ก, ๐ฆ1 ) โ ๐(๐ก, ๐ฆ2 )| โค ๐ด|๐ฆ1 โ ๐ฆ2 |, ๐ด > 0, where ๐ : N๐,๐ ร R โ R and ๐ฆ : N๐,๐ โ R. Then, system (34) has a unique solution of the form
(36)
Then, for arbitrary ๐ฅ1 , ๐ฅ2 โ ๐ and ๐ก โ N๐,๐ , we have by assumption that ๓ตจ๓ตจ ๓ตจ ๓ตจ๓ตจ(๐๐ฅ1 ) (๐ก) โ (๐๐ฅ2 ) (๐ก)๓ตจ๓ตจ๓ตจ ๓ตจ๓ตจ ๓ตจ๓ตจ โ๐ผ = ๓ตจ๓ตจ๓ตจCF๐ โ [๐ (๐ก, ๐ฅ1 (๐ก)) โ ๐ (๐ก, ๐ฅ2 (๐ก))]๓ตจ๓ตจ๓ตจ (37) ๓ตจ ๓ตจ โค ๐ด(
such that (๐ด/๐ต(๐ผโ1))((2โ๐ผ)(๐โ๐)+(๐ผโ1)(๐โ๐)2 /2) < 1, and |๐(๐ก, ๐ฆ1 )โ๐(๐ก, ๐ฆ2 )| โค ๐ด|๐ฆ1 โ๐ฆ2 |, ๐ด > 0, where ๐ : N๐,๐ รR โ R and ๐ฆ : N๐,๐ โ R. Then, system (34) has a unique solution of the form โ๐ผ
๐ผ
(๐ถ๐น๐ถ๐โ ๐ฆ) (๐ก) = ๐ (๐ก, ๐ฆ (๐ก)) ,
๐ฆ (๐ก) = ๐ +
(39)
1 โ ๐ผ ๐ผ (๐ โ ๐) ๓ตฉ๓ตฉ ๓ตฉ + ) ๓ตฉ๓ตฉ๐ฅ1 โ ๐ฅ2 ๓ตฉ๓ตฉ๓ตฉ , ๐ต (๐ผ) ๐ต (๐ผ)
and hence ๐ is a contraction. By Banach fixed point theorem, there exists a unique ๐ฅ โ ๐ such that ๐๐ฅ = ๐ฅ and hence the proof is complete. Remark 15. Similar existence and uniqueness theorems can be proved for system (34) with higher order by making use of
+
๐ก 2โ๐ผ โ ๐ (๐ , ๐ฆ (๐ )) ๐ต (๐ผ โ 1) ๐ =๐+1
(40)
๐ผโ1 ( โโ2 ๐ (โ
, ๐ฆ (โ
)) (๐ก)) . ๐ต (๐ผ โ 1) ๐ โ๐ผ
Proof. If we apply CF๐ โ to system (39) and make use of Proposition 10 with ๐ฝ = ๐ผ โ 1 then we reach at the ๐ผ representation (40). Conversely, if we apply CFR๐ โ , make use of Proposition 10 and by noting that CFR ๐ผ ๐โ
๐ฝ
= CFR๐ โ โ๐ก ๐ = 0,
(41)
we obtain system (39). Hence, ๐ฆ(๐ก) satisfies system (39) if and only if it satisfies (40). Let ๐ = {๐ฅ : max๐กโN๐,๐ |๐ฅ(๐ก)| < โ} be the Banach space endowed with the norm โ๐ฅโ = max๐กโN๐,๐ |๐ฅ(๐ก)|. On ๐ define the linear operator โ๐ผ
(๐๐ฅ) (๐ก) = ๐ + CF๐ โ ๐ (๐ก, ๐ฅ (๐ก)) .
(42)
Then, for arbitrary ๐ฅ1 , ๐ฅ2 โ ๐ and ๐ก โ N๐,๐ , we have by assumption that ๓ตจ๓ตจ ๓ตจ ๓ตจ๓ตจ(๐๐ฅ1 ) (๐ก) โ (๐๐ฅ2 ) (๐ก)๓ตจ๓ตจ๓ตจ ๓ตจ๓ตจ ๓ตจ๓ตจ โ๐ผ = ๓ตจ๓ตจ๓ตจCF๐ โ [๐ (๐ก, ๐ฅ1 (๐ก)) โ ๐ (๐ก, ๐ฅ2 (๐ก))]๓ตจ๓ตจ๓ตจ ๓ตจ ๓ตจ (43) ๐ด (๐ผ โ 1) (๐ โ ๐)2 โค ((2 โ ๐ผ) (๐ โ ๐) + ) ๐ต (๐ผ โ 1) 2 ๓ตฉ ๓ตฉ โ
๓ตฉ๓ตฉ๓ตฉ๐ฅ1 โ ๐ฅ2 ๓ตฉ๓ตฉ๓ตฉ , and hence ๐ is a contraction. By Banach Contraction Principle, there exists a unique ๐ฅ โ ๐ such that ๐๐ฅ = ๐ฅ and hence the proof is complete.
6
Discrete Dynamics in Nature and Society Lemma 18. Given that ๐ โก ๐ (mod 1), the Green function ๐บ(๐ก, ๐ ) defined in Lemma 17 has the following properties:
5. The Lyapunov Inequality for the CFR Difference Boundary Value Problem In this section, we prove a Lyapunov inequality for a CFR boundary value difference problem of order 2 < ๐ผ โค 3. Consider the boundary value problem ๐ผ (CFR๐ โ ๐ฆ) (๐ก)
๐
+ ๐ (๐ก) ๐ฆ (๐ก) = 0,
2 < ๐ผ โค 3, ๐ก โ N๐+1,๐โ1 , ๐ฆ (๐) = ๐ฆ (๐) = 0,
(44)
where, ๐ฆ๐ (๐ก) = ๐ฆ(๐(๐ก)) = ๐ฆ(๐ก โ 1). Lemma 17. ๐ฆ(๐ก) is a solution of the boundary value problem (44) if and only if it satisfies the equation ๐
๐ฆ (๐ก) = โ ๐บ (๐ก, ๐ ) ๐ (๐ , ๐ฆ (๐ )) , where ๐บ (๐ก, ๐ )
๐ (๐ก, ๐ฆ (๐ก)) = =
๐ถ๐น ๐ฝ ๐โ
๐ก + 1 โค ๐ , ๐ก, ๐ โ N๐,๐ , ๐ โ 1 โค ๐ก, ๐ก, ๐ โ N๐,๐ ,
(46)
(๐ (โ
) ๐ฆ๐ (โ
)) (๐ก)
๐ฝ 1โ๐ฝ ๐ (๐ก) ๐ฆ๐ (๐ก) + ( โโ1 ๐ (โ
) ๐ฆ๐ (โ
)) (๐ก) , ๐ฝ = ๐ผ โ 2. ๐ต (๐ฝ) ๐ต (๐ฝ) ๐ โ๐ผ
Proof. Apply the integral CF๐ โ to (44) and make use of Definition 6 and Proposition 10 with ๐ = 2 and ๐ฝ = ๐ผ โ 2 to reach ๐ฆ (๐ก) = ๐1 + ๐2 (๐ก โ ๐) โ (๐ โโ2 ๐ (โ
, ๐ฆ (โ
))) (๐ก) ๐ก
= ๐1 + ๐2 (๐ก โ ๐) โ โ (๐ก โ ๐ (๐ )) ๐ (๐ , ๐ฆ (๐ )) .
(47)
๐ =๐+1
The condition ๐ฆ(๐) = 0 implies that ๐1 = 0 and the condition ๐ฆ(๐) = 0 implies that ๐2 = (1/(๐ โ ๐)) โ๐๐ =๐+1 (๐ โ ๐(๐ ))๐(๐ , ๐ฆ(๐ )), and hence ๐ฆ (๐ก) =
๐กโ๐ ๐ โ (๐ โ ๐ (๐ )) ๐ (๐ , ๐ฆ (๐ )) ๐ โ ๐ ๐ =๐+1 ๐ก
(48)
โ โ (๐ก โ ๐ (๐ )) ๐ (๐ ) ๐ (๐ , ๐ฆ (๐ )) ๐๐ . ๐ =๐+1
Then, the result follows by splitting the summation โ (๐ โ ๐ (๐ )) ๐ (๐ , ๐ฆ (๐ )) ๐ก
๐
+ โ (๐ โ ๐ (๐ )) ๐ (๐ , ๐ฆ (๐ )) . ๐ =๐ก+1
๐โ๐ (๐ + ๐ + 2) { )= { {๐ ( 2 4 ={ + ๐ + 3) โ ๐)2 โ 1 (๐ (๐ { {๐ ( )= 2 4 (๐ โ ๐) {
if ๐ โ ๐ is even (50) if ๐ โ ๐ is odd.
Hence, in either cases max๐ โN๐,๐ ๐(๐ ) โค (๐ โ ๐)/4. Proof. (1) It is clear that ๐1 (๐ก, ๐ ) = (๐ก โ ๐)(๐ โ ๐(๐ ))/(๐ โ ๐) โฅ 0. Regarding the part ๐2 (๐ก, ๐ ) = ((๐กโ๐)(๐โ๐(๐ ))/(๐โ๐)โ(๐กโ๐(๐ ))) we see that (๐ก โ ๐(๐ )) = ((๐ก โ ๐)/(๐ โ ๐))(๐ โ (๐ + (๐(๐ ) โ ๐)(๐ โ ๐)/(๐ก โ ๐))) and that ๐ + (๐(๐ ) โ ๐)(๐ โ ๐)/(๐ก โ ๐) โฅ ๐(๐ ) if and only if ๐(๐ )(๐ โ ๐ก) + ๐(๐ก โ ๐) โฅ 0 if and only if ๐ โฅ ๐ + 1. Hence, we conclude that ๐2 (๐ก, ๐ ) โฅ 0 as well. (2) Clearly, ๐1 (๐ก, ๐ ) is an increasing function in ๐ก. Applying โ to ๐2 with respect to ๐ก for every fixed ๐ โฅ ๐ + 1 we see that โ๐ก ๐(๐ก, ๐ ) = (๐ โ ๐)โ1 (๐ โ ๐(๐ )) and hence ๐2 is a decreasing function in ๐ก. (3) Let ๐(๐ ) = ๐บ(๐(๐ ), ๐ ) = (๐(๐ ) โ ๐)(๐ โ ๐(๐ ))/(๐ โ ๐). Then, ๐ + ๐ โ 2๐ + 3 (51) = 0, ๐โ๐ if ๐ = (๐ + ๐ + 3)/2 and hence for ๐, ๐ โ N, ๐ attains its maximum at ๐ 1 = (๐ + ๐ + 3)/2 if ๐ + ๐ (or ๐ โ ๐) is odd and at (๐ + ๐ + 2)/2 if ๐ + ๐ (or ๐ โ ๐) is even. More generally, if ๐ and ๐ are such that ๐ โก ๐ (mod 1), we see that ๐ 1 โก ๐ (mod 1) if ๐ โ ๐ is odd and ๐ 2 โก ๐ (mod 1) if ๐ โ ๐ is even. Finally, ๐(๐ 1 ) = ((๐ โ ๐)2 โ 1)/4(๐ โ ๐) and ๐(๐ 2 ) = (๐ โ ๐)/4. (โ๐) (๐ ) =
In next lemma, we estimate ๐(๐ก, ๐ฆ(๐ก)) for a function ๐ฆ โ ๐ต[N๐,๐ ], the Banach space of all Banach-valued finite sequences on N๐,๐ with โ๐ฆโ = max๐กโN๐,๐ |๐ฆ(๐ก)|, where | โ
| is the norm in the Banach space. Lemma 19. For ๐ฆ โ ๐ต[N๐,๐ ] and 2 < ๐ผ โค 3, ๐ฝ = ๐ผ โ 2, we have, for any ๐ก โ N๐,๐ , ๓ตจ๓ตจ ๓ตจ ๓ตฉ ๓ตฉ (52) ๓ตจ๓ตจ๐ (๐ก, ๐ฆ (๐ก))๓ตจ๓ตจ๓ตจ โค ๐
(๐ก) ๓ตฉ๓ตฉ๓ตฉ๐ฆ๓ตฉ๓ตฉ๓ตฉ ,
๐
(๐ก) = [
๐ =๐+1
๐ =๐+1
max ๐ (๐ )
๐ โN๐,๐
where
๐
= โ (๐ โ ๐ (๐ )) ๐ (๐ , ๐ฆ (๐ ))
(3) ๐(๐ ) = ๐บ(๐(๐ ), ๐ ) = (๐(๐ ) โ ๐)(๐ โ ๐(๐ ))/(๐ โ ๐) has a unique maximum, given by
(45)
๐ =๐+1
(๐ก โ ๐) (๐ โ ๐ (๐ )) { , { { ๐โ๐ ={ { {( (๐ก โ ๐) (๐ โ ๐ (๐ )) โ (๐ก โ ๐ (๐ ))) , ๐โ๐ {
(1) ๐บ(๐ก, ๐ ) โฅ 0 for all ๐ก, ๐ โ N๐,๐ . (2) max๐กโN๐,๐ ๐บ(๐ก, ๐ ) = ๐บ(๐(๐ ), ๐ ) for ๐ โ N๐+1,๐โ1 .
(49)
๐ก 3 โ ๐ผ ๓ตจ๓ตจ ๐ผโ2 ๓ตจ ๓ตจ ๓ตจ โ ๓ตจ๓ตจ๓ตจ๐ (๐ )๓ตจ๓ตจ๓ตจ] . ๓ตจ๓ตจ๐ (๐ก)๓ตจ๓ตจ๓ตจ + ๐ต (๐ผ โ 2) ๐ต (๐ผ โ 2) ๐ =๐+1
(53)
Theorem 20 (the CFR fractional difference Lyapunov inequality). If the boundary value problem (44) has a nontrivial solution, where ๐(๐ก) is a real-valued bounded function on N๐,๐ , then ๐
โ ๐
(๐ ) > ๐ =๐+1
4 . ๐โ๐
(54)
Discrete Dynamics in Nature and Society
7
Proof. Assume ๐ฆ โ ๐ = ๐ต[N๐,๐ ] is a nontrivial solution of the boundary value problem (44). By Lemma 17, ๐ฆ must satisfy ๐
๐ฆ (๐ก) = โ ๐บ (๐ก, ๐ ) ๐ (๐ , ๐ฆ (๐ )) .
(55)
๐ =๐+1
Then, by using the properties of the Green function ๐บ(๐ก, ๐ ) proved in Lemmas 18 and 19, we come to the conclusion that ๐ ๓ตฉ ๓ตฉ ๓ตฉ๓ตฉ ๓ตฉ๓ตฉ ๐ โ ๐ โ ๐
(๐ ) ๓ตฉ๓ตฉ๓ตฉ๐ฆ๓ตฉ๓ตฉ๓ตฉ ๓ตฉ๓ตฉ๐ฆ๓ตฉ๓ตฉ < 4 ๐ =๐+1
(56)
from which (54) follows. Remark 21. Note that if ๐ผ โ 2+ , then ๐
(๐ก) tends to |๐(๐ก)| and hence we obtain the classical nabla discrete version of the Lyapunov inequality (2). For the sake of more comparisons of Lyapunov inequalities on time scales we refer to [40]. Example 22. Consider the following CFR Sturm-Liouville difference eigenvalue problem (SLDEP) of order 2 < ๐ผ โค 3
๐ก โ N1,๐โ1 , ๐ โ N, ๐ฆ (0) = ๐ฆ (๐) = 0.
(57)
If ๐ is an eigenvalue of (57), then by Theorem 20 with ๐(๐ก) = ๐, we have ๐ (๐ก) = [
3โ๐ผ ๐ผโ2 ( โโ1 |๐|) (๐ก)] |๐| + ๐ต (๐ผ โ 2) ๐ต (๐ผ โ 2) 0
= |๐| [
3โ๐ผ ๐ผโ2 + ๐ก] . ๐ต (๐ผ โ 2) ๐ต (๐ผ โ 2)
๐
๐ =1
๐ (3 โ ๐ผ) ๐2 (๐ผ โ 2) 4 + ]> . ๐ต (๐ผ โ 2) 2๐ต (๐ผ โ 2) ๐
(58)
(59)
Notice that the limiting case ๐ผ โ 2+ implies that |๐| > 4/๐2 which is the lower bound for the eigenvalues of the ordinary difference eigenvalue problem: (โ2 ๐ฆ) (๐ก) + ๐๐ฆ๐ (๐ก) = 0, ๐ก โ N1,๐โ1 , ๐ฆ (0) = ๐ฆ (๐) = 0.
The authors declare that they have no conflicts of interest.
The first author would like to thank the Research and Translation Center in Prince Sultan University for continuous encouragement and support.
References
Hence, we must have โ๐ (๐ ) = |๐| [
Conflicts of Interest
Acknowledgments
๐ผ
(CFR0 โ ๐ฆ) (๐ก) + ๐๐ฆ๐ (๐ก) = 0,
achieved. To set up the basic concepts, we proved existence and uniqueness theorems by means of Banach fixed point theorem for initial value problems in the frame of CFC and CFR fractional differences. We have come to the conclusion that the condition ๐(๐, ๐ฆ(๐)) = 0 is necessary to guarantee the existence of solution and hence fractional linear difference initial value problem with constant coefficients results in the trivial solution unless the order is positive integer. We used our extension to arbitrary order to prove a Lyapunov type inequality for a CFR boundary value problem of order 2 < ๐ผ โค 3 and then obtain the classical ordinary case when ๐ผ tends to 2 from right. This proves different behavior from the classical fractional difference case, where the Lyapunov inequality was proved for a fractional difference boundary problem of order 1 < ๐ผ โค 2 and the classical ordinary case was then recovered when ๐ผ tends to 2 from left.
(60)
6. Conclusions Fractional differences and their correspondent fractional sum operators are of importance in discrete modeling of various problems in science. We extended the fractional difference calculus whose difference operators depend on a discrete exponential function kernel to arbitrary positive order. The correspondent arbitrary order fractional sum operators have been defined as well and applied to solve fractional initial and boundary value difference problems. The extension for right fractional differences and sums is also
[1] M. A. Hajji, Q. M. Al-Mdallal, and F. M. Allan, โAn efficient algorithm for solving higher-order fractional Sturm-Liouville eigenvalue problems,โ Journal of Computational Physics, vol. 272, pp. 550โ558, 2014. [2] Q. M. Al-Mdallal and M. A. Hajji, โA convergent algorithm for solving higher-order nonlinear fractional boundary value problems,โ Fractional Calculus and Applied Analysis, vol. 18, no. 6, pp. 1423โ1440, 2015. [3] Q. M. Al-Mdallal, โOn the numerical solution of fractional Sturm-Liouville problems,โ International Journal of Computer Mathematics, vol. 87, no. 12, pp. 2837โ2845, 2010. [4] A. H. Bhrawy, T. M. Taha, and J. A. Machado, โA review of operational matrices and spectral techniques for fractional calculus,โ Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, vol. 81, no. 3, pp. 1023โ1052, 2015. [5] F. Wu and J. Liu, โDiscrete fractional creep model of salt rock,โ J. Comput. Complex. Appl, vol. 2, pp. 1โ6, 2016. [6] R. Abu-Saris and Q. Al-Mdallal, โOn the asymptotic stability of linear system of fractional-order difference equations,โ Fractional Calculus and Applied Analysis. An International Journal for Theory and Applications, vol. 16, no. 3, pp. 613โ629, 2013. [7] T. Abdeljawad and D. Baleanu, โOn fractional derivatives with exponential kernel and their discrete versions,โ Journal of Reports in Mathematical Physics, https://arxiv.org/abs/1606 .07958. [8] M. Caputo and M. Fabrizio, โA new definition of fractional derivative without singular kernal,โ Progress in Fractional Differentiation and Applications, vol. 1, no. 2, pp. 73โ85, 2015.
8 [9] J. Losada and J. J. Nieto, โProperties of a new fractional derivative without singular kernal,โ Progress in Fractional Differentiation and Applications, vol. 1, no. 2, pp. 87โ92, 2015. [10] T. Abdeljawad and D. Baleanu, โMonotonicity results for fractional difference operators with discrete exponential kernels,โ Advances in Difference Equations, vol. 2017, Article ID 78, 2017. [11] A. Atangana and D. Baleanu, โNew fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model,โ Thermal Science, vol. 20, no. 2, pp. 763โ 769, 2016. [12] T. Abdeljawad and D. Baleanu, โIntegration by parts and its applications of a new nonlocal fractional derivative with MittagLeffler nonsingular kernel,โ The Journal of Nonlinear Sciences and Applications, vol. 10, no. 03, pp. 1098โ1107, 2017. [13] T. Abdeljawad and D. Baleanu, โDiscrete fractional differences with nonsingular discrete Mittag-Leffler kernels,โ Advances in Difference Equations, vol. 2016, Article ID 232, 2016. [14] T. Abdeljawad and D. Baleanu, โMonotonicity analysis of a nabla discrete fractional operator with discrete Mittag-Leffler kernel,โ Chaos, Solitons & Fractals, 2017. [15] R. A. Ferreira, โSome discrete fractional Lyapunov-type inequalities,โ Fractional Differential Calculus, vol. 5, no. 1, pp. 87โ92, 2015. [16] R. A. Ferreira, โA Lyapunov-type inequality for a fractional boundary value problem,โ Fractional Calculus and Applied Analysis. An International Journal for Theory and Applications, vol. 16, no. 4, pp. 978โ984, 2013. [17] A. Chidouh and D. F. Torres, โA generalized Lyapunovโs inequality for a fractional boundary value problem,โ Journal of Computational and Applied Mathematics, vol. 312, pp. 192โ197, 2017. [18] M. Jleli and B. Samet, โLyapunov-type inequalities for fractional boundary-value problems,โ Electronic Journal of Differential Equations, vol. 2015, no. 88, 11 pages, 2015. [19] D. OโRegan and B. Samet, โLyapunov-type inequalities for a class of fractional differential equations,โ Journal of Inequalities and Applications, vol. 2015, no. 247, 10 pages, 2015. [20] J. Rong and C. Bai, โLyapunov-type inequality for a fractional differential equation with fractional boundary conditions,โ Advances in Difference Equations, vol. 2015, no. 82, 10 pages, 2015. [21] M. Jleli and B. Samet, โA Lyapunov-type inequality for a fractional ๐-difference boundary value problem,โ Journal of Nonlinear Science and its Applications. JNSA, vol. 9, no. 5, pp. 1965โ1976, 2016. [22] H. Esser, โStability inequalities for discrete nonlinear two-point boundary value problems,โ Applicable Analysis. An International Journal, vol. 10, no. 2, pp. 137โ162, 1980. [23] N. G. Abuj and D. B. Pachpatte, โLyapunov type inequality for fractional differential equation with k-Prabhakar derivative,โ https://arxiv.org/abs/1702.01562. [24] D. B. Pachpatte, N. G. Abuj, and A. D. Khandagale, โLyapunov type inequality for hybrid fractional differential equation with Prabhakar Derivative,โ https://arxiv.org/abs/1609.03027. [25] X. Yang, โOn Liapunov-type inequality for certain higher-order differential equations,โ Applied Mathematics and Computation, vol. 134, no. 2-3, pp. 307โ317, 2003. [26] A. n. Tiryaki, โRecent developments of Lyapunov-type inequalities,โ Advances in Dynamical Systems and Applications, vol. 5, no. 2, pp. 231โ248, 2010.
Discrete Dynamics in Nature and Society [27] N. Parhi and S. Panigrahi, โLiapunov-type inequality for higher order differential equations,โ Mathematica Slovaca, vol. 52, no. 1, pp. 31โ46, 2002. [28] J. X. Cui, X. Huang, and M. C. Hou, โLyapunov-type inequality for fractional order difference equations,โ Global Journal of Science Frontier Research, vol. 16, no. 1, 2016. [29] T. Abdeljawad, โA Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel,โ Journal of Inequalities and Applications, 2017:130 pages, 2017. [30] A. M. Lyapunov, โProbleme gยดenยดeral de la stabilitยดe du mouvment,โ Annales de la Facultยดe des Sciences de Toulouse, vol. 2, pp. 227โ247, 1907, Reprinted in: Ann. Math. Studies, no. 17, Princeton, 1947. [31] S. S. Cheng, โA discrete analogue of the inequality of Lyapunov,โ Hokkaido Mathematical Journal, vol. 12, no. 1, pp. 105โ112, 1983. [32] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993. [33] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. [34] A. Kilbas, M. H. Srivastava, and J. J. Trujillo, Theory and Application of Fractional Differential Equations, vol. 204 of North Holland Mathematics Studies, 2006. [35] F. Bozkurt, T. Abdeljawad, and M. A. Hajji, โStability analysis of a fractional order differential equatin model of a brain tumor growth depending on the density,โ Applied and Computational Mathematics. An International Journal, vol. 14, no. 1, pp. 50โ62, 2015. [36] C. Goodrich and A. C. Peterson, Discrete Fractional Calculus, Springer International, New York, NY, USA, 2015. [37] T. Abdeljawad, โOn delta and nabla Caputo fractional differences and dual identities,โ Discrete Dynamics in Nature and Society, vol. 2013, Article ID 406910, 12 pages, 2013. [38] T. Abdeljawad, โDual identities in fractional difference calculus within Riemann,โ Advances in Difference Equations, vol. 2013, no. 36, 2013. [39] T. Abdeljawad and D. F. M. Torres, โSymmetric duality for left and right Riemann-Liouville and Caputo fractional differences,โ Arab Journal of Mathematical Sciences, vol. 23, no. 2, pp. 157โ172, 2017. [40] M. Bohner and A. Peterson, Dynamic equations on time scales, Springer, Berlin, Germany, 2001.
Advances in
Operations Research Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Journal of
Applied Mathematics
Algebra
Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Journal of
Probability and Statistics Volume 2014
The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at https://www.hindawi.com International Journal of
Advances in
Combinatorics Hindawi Publishing Corporation http://www.hindawi.com
Mathematical Physics Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
International Journal of Mathematics and Mathematical Sciences
Mathematical Problems in Engineering
Journal of
Mathematics Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
#HRBQDSDฤฎ,@SGDL@SHBR
Journal of
Volume 201
Hindawi Publishing Corporation http://www.hindawi.com
Discrete Dynamics in Nature and Society
Journal of
Function Spaces Hindawi Publishing Corporation http://www.hindawi.com
Abstract and Applied Analysis
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
International Journal of
Journal of
Stochastic Analysis
Optimization
Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014